An important regression in finance is the regression of the excess return of an asset or portfolio or fund on the excess return of a benchmark.
Excess return = return minus risk-free rate
\[r-r_{f}=\alpha + \beta(r_b-r_f)+\epsilon\]
\(r\)=return
\(r_{f}\)=risk-free return
\(r_{b}\)=benchmark return
\(\epsilon\)=zero-mean risk uncorrelated with \(r_{b}\)
For example, asset = stock and benchmark = stock market return
Meaning of \(\beta\)
Beta answers this question:
if the benchmark is up 1%, how much do we expect the asset to be up, all else equal?
- If \(\beta\)=2, we expect the asset to be up 2%
- If \(\beta\)=0.5, we expect the asset to be up 0.5%
4A
Meaning of \(\alpha\)
Alpha answers this question:
If I start by holding the benchmark, can I improve mean-variance efficiency by investing something in the asset?
- The answer is “yes” if and only if \(\alpha\) > 0
- If \(\alpha\) < 0, mean-variance efficiency can be improved by shorting the asset.
4B
What Sharpe Ratio is Needed for > 0?
Performance is often measured by Sharpe ratio
= reward to risk ratio
= risk premium / std dev
\(\alpha > 0\) if and only if
Sharpe ratio > Sharpe ratio of benchmark \(\times\) correlation
Low correlation \(\rightarrow\) \(\alpha > 0\) with low Sharpe ratio.
4C